XI-A-ST: A supported Einstein–scalar sheet on the prescribed Enter spacetime Signed action, separated Noether identities, an empty fixed-input Einstein jet, and complete finite reports

We give a local Grade S transfer of Cheung's XI-A Einstein–scalar sheet to the regular I-B conical product. The host geometry is prescribed, while all analytical equations remain on an independently declared Minkowski completion. The transfer retains the unswitched Einstein–Hilbert action, the minimally switched scalar action, the distinction between invariant and gauge-fixed Noether identities, and tensor-density covariance of the invariant metric Euler operator.

The principal obstruction is inherited, not newly claimed: for the specified free scalar input ${\varphi }_m(t)=\sin (mt)/m$, with massless value $t$, the fixed-input Einstein jet is empty for every nonzero compact switch in the frozen class. Its detector is the first source-compatibility coefficient $({\partial }_{\mu }g){T_0}^{\mu \nu }[{\varphi }_m]$. A compact bump gives a strictly positive margin, whereas the vacuum retains a nonempty jet. This distinguishes a fixed-input incompatibility from a failure of Einstein theory or of gauge-fixed wave solvability.

The contribution of this sheet is the supported transfer and its complete reporting interface, not a new gravitational covariance theorem. We identify two essential locality qualifications: the switching detector is not a plateau detector, and global completion-jet emptiness must not be renamed emptiness of every plateau-local solution space. A finite supplied catalogue is reported by destructive Enter, with surviving registers and arbitrary untouched references retained. An explicit common-interface probability encoding has an exact diamond-distance formula for rational table approximations. Finite-probe coverage and an invisible conserved-perturbation construction prevent a successful finite table from being mistaken for a distributional identity.

The potential value is an auditable separation of action identities, solution compatibility, local support, and finite readout. The fixed positive ramp width is not a worldsheet thickness. No host reconstruction, field-acquisition map, gravitational propagator, quantum multiplet, tree, BRST construction, vacuum-energy calculation, or characteristic quantum coefficient is produced.

release{ The classical XI-A sheet is retained on the supported completion chart. The specified completion Einstein jet is proved obstructed, with the same compact detector whenever that detector fits the retained laboratory.

The finite report instrument is constructed for supplied evaluations. It does not acquire the continuum fields, source the host, or turn a report of incompatibility into a metric solution.

X-A-ST's relational intertwiner, scalar covariance, and edge-zone theorem are cited, not recounted.}

XI-A fixes one classical Einstein–scalar action, separates its invariant and gauge-fixed identities, and identifies an empty Einstein fiber for a named free scalar input CheungXIA. The present sheet retains those objects. It changes their supported host interpretation and supplies a finite Enter report interface.

The action is not obtained from the conical metric. Conversely, imposing an equation on a formal analysis metric does not update that conical metric. These two statements are part of the model specification, not consequences of a field equation.

The results attributed below to XI-A are supported restatements of its explicitly named formulas and theorems. Short variational, sign, and chart checks fix the transfer. They are not offered as new discoveries of the underlying Einstein–scalar identities.

The broader distinction between classical gauge symmetry, gauge fixing, and subsequent perturbative BV quantization is also explicit in the external literature BFR. No BV construction is imported into this sheet.

Cheung's Enter calculus uses a strictly linear treatment of quantum names and destructive computational-basis readout. Its operational characterization of measurement is class-relative; it is not a characterization of every possible continuum field instrument CheungEnter,CheungI.

The accompanying philosophy essay distinguishes the classical record returned by destructive readout from a quantum-return recovery map CheungPhilosophy. The relevant questions are:

Here the supplied objects are finitely many classical evaluations and their declared encodings. The output contains a classical code and the actual unmeasured finite residual.

In particular,

The complete finite instrument is proved in Section sec:instrument; no general measurement uniqueness theorem is needed for that matrix calculation.

The gravitational action, its signs, its Noether identity, and the fixed-input obstruction belong to XI-A. The host and destructive clock interface belong to the I-B/X-A-ST framework CheungIB,CheungXAST.

The present assembly makes explicit:

1. the local tensor-density transfer with the two signature conventions kept separate;

2. the distinction between a switching-region detector and a plateau-local equation problem;

3. a complete finite reporting channel with a specified rational-approximation metric;

4. the coverage requirements that separate a finite report from an identity on an unrestricted residual space.

No priority claim is made for the general variational, linear-algebraic, or channel principles used in this assembly.

Retain the regular conical product

with metric

and future orientation given by increasing $T$. Its completion has the worldsheet

The defect is a worldsheet, not a particle worldline.

Retain a regular developing chart

from an open set $V$ in the declared completion

The signature and time-calibration ledger is

Retain the observation region $\mathcal{O}$ and the supported shadow $D_{\ast }⋐V$ declared for this laboratory instance.

This is the I-B host with the X-A-ST completion convention CheungIB,CheungXAST. The notation $V$ makes the chart domain distinct from its host image $\mathcal{L}$.

The host declarations and the non-global-transfer boundary are copied from I-B/X-A-ST rather than reproved CheungIB,CheungXAST. The three-storage-ledger discipline is the one used in X-L-ST CheungXLST. It does not import that sheet's action or free normalization.

Equation (eq:signature) does not assert that a real coordinate transformation changes the signature of a bilinear form. It specifies a sign-reversed analytical comparison metric on the developed chart. All analytical contractions below use $\eta$, or the formal metric $q$, not the host metric $G$.

The fixed $\epsilon >0$ belongs to the ramp prescription. It neither thickens $\mathcal{W}$ nor defines a smooth core replacing the cone. No width limit is taken.

Let $x$ be the developed coordinates and $y$ an orientation-preserving coordinate system on the host laboratory. If $u^{\mu \nu }$ is a contravariant rank-two density of weight one, its transported components are

Compactly supported variational pairings, density identities, and nonzero compact-test pairings are unchanged by the transport (eq:passivedensity), provided all background tensors and test tensors are transported consistently.

This local rule supplies no global conical Cauchy inverse and no field equation on the worldsheet.

Proof. A covariant variation transforms with two inverse Jacobian factors. They cancel the two displayed tensor factors in (eq:passivedensity); the remaining determinant is exactly the change-of-variables factor in the integral. Thus the variational pairing is unchanged.

An invertible change of coordinates maps zero distributions to zero distributions and preserves a detecting pairing after the test is transported. The calculation is confined to $V$ and makes no assertion outside the regular chart.

In particular, no additional minus sign is inserted into the XI-A action because the host convention is $(-,+,+,+)$. That action is still the analysis action with background $\eta$.

Work on

as subsets of $\hat{M}$. Every coefficient field is real and smooth on a neighborhood of the closed slab.

Fix

Retain $0<\epsilon <T$ and

where

Choose the retained plateau domain

The admitted parameters are compactly supported in $U$.

The plateau group ${\mathcal{G}}_P$ consists of the corresponding orientation-preserving compact flows. For order bookkeeping, use genuine flows generated by $\kappa \xi +{\kappa }^2\zeta$, with $\xi ,\zeta \in {C_c}^{\infty }(U;T{M}_T)$. They preserve $g$ and ${\sigma }_{\epsilon }$ and are the identity near $\Sigma$.

The independent classical metric variable is one symmetric tensor $\gamma$, with

The scalar is $\varphi$, and

The formal metric $q$ is not $G$. Evaluating $\gamma$ later on a formal coefficient family does not introduce a second fundamental metric field.

All expansions are coefficientwise in $\kappa$. They are not claims about convergence of a numerical perturbation series.

The following remain the relational baseline CheungXAST (:edge, :main):

1. the complete second-order coordinate intertwiner;

2. off-shell invariance of the label-smeared relational scalar;

3. covariance of the switched scalar solution with its specified Cauchy data;

4. the strictly negative edge-zone preflight defect,

   $$-{{\int }_0}^{\epsilon }{\sigma }_{\epsilon }^{\prime }(s){\alpha }_{\mathrm{e}}(s)\mathrm{d}s<0$$

   on its named detecting data.

These results are not proved again here. The profile ${\alpha }_{\mathrm{e}}$ in this documentary formula is unrelated to the cone opening ${\alpha }_c$.

In particular, contracting the Noether identity below on a plateau parameter is not presented as a new Route II theorem for that observable.

X-B, X-C, and X-D remain non-inputs of the classical action sheet, as in XI-A CheungXIA. The X-E normalization is not substituted for the one below.

Write ${\rho }_q=\sqrt{-\det q}$. Use the curvature convention

The retained XI-A normalization is

There is no cosmological-constant term and no gravitational switch in (eq:EH) CheungXIA.

The ramp regulates the inherited relational coordinate prescription. It does not multiply the Einstein–Hilbert density.

Actions mean variational functionals under compact interior variations and compact differences. Their values need not be finite numbers on arbitrary noncompact smooth configurations. No boundary variation of the metric is taken.

Define

The single gauge-fixing term is

Its coefficient is frozen. No gauge-parameter family is used.

The bulk quadratic coefficient of (eq:EH), modulo a total divergence, is

\begin{split}
 S_{\mathrm{FP}}[\gamma]=\frac12\int_{M_T}\Big[
 &\partial_\lambda\gamma_{\mu\nu}
                       \partial^\lambda\gamma^{\mu\nu}
 -2(\partial_\mu\gamma^{\mu\nu})
                       (\partial^\rho\gamma_{\rho\nu})\\
 &+2(\partial_\mu\gamma^{\mu\nu})\partial_\nu\tau
 -\partial_\lambda\tau\partial^\lambda\tau
 \Big]\mathrm d^4x.
 \label{eq:FP}
\end{split}
Its Euler derivative is \(2G_{\mathrm{lin}}^{\mu\nu}(\gamma)\), where
\begin{equation}
 G_{\mathrm{lin}}^{\mu\nu}(\gamma)
 =
 -\frac12\Box\bar\gamma^{\mu\nu}
 +\partial^{(\mu}D^{\nu)}
 -\frac12\eta^{\mu\nu}\partial_\rho D^\rho.
 \label{eq:Glin}

Parentheses include the factor $1/2$.

The separate gauge-fixing derivative is

Consequently the quadratic gauge-fixed operator is

Proof. These are XI-A :FP, :Glin, and :egf CheungXIA. The relevant action variation is

Since $\delta q=\kappa \delta \gamma$, its leading Euler coefficient is $2G_{\mathrm{lin}}$. Compact variation of (eq:gf) gives (eq:egf); addition gives (eq:quadraticoperator). Lemma lem:transfer transports these variational pairings without changing the action sign.

In particular, this handoff is $-\square {\mathsf{P}}_{\mathrm{tr}}$, not X-E's differently normalized metric operator. A shared de Donder name does not identify them.

two retained sign diagnostics

For

the Fierz–Pauli density is

For the auxiliary static unswitched comparison with $T^{00}=\rho \ge 0$, $T^{0i}=T^{ij}=0$, where $\rho$ is smooth, compactly supported, and nonzero, put $H=q-\eta$. The linear equation has the comparison

with

Thus ${\kappa }^2=32\pi G_N$ has the attractive sign.

Proof. These are XI-A :TT and :Newton CheungXIA. Raising both spatial indices in (eq:TT) contributes two minus signs, so the time-derivative coefficient remains positive. In (eq:Newtonmetric), ${\bar{H}}_{00}=4{\Phi }_N$, ${\bar{H}}_{ij}=0$, and $D(H)=0$. Equation (eq:Glin) gives ${G_{\mathrm{lin}}}^{00}(H)=2\Delta {\Phi }_N$. The retained unswitched equation is $G^{\mu \nu }={\kappa }^2{T}^{\mu \nu }/4$, giving (eq:Newton). Neither calculation inserts $G$ in place of $q$.

The Newtonian source is an auxiliary static comparator. It is not a solution of the compact-time switched system. Neither sign diagnostic computes vacuum energy.

metric coupling and the actual scalar switch

Let $p_{\mu }={\partial }_{\mu }\varphi$, and put

Write $T_0=T_{\eta }$.

The covariant metric variation is

Set

where

The matter vertices are therefore

There is one factor $g$, not $g^2$, in the second coefficient.

Define ${H_n}^{\mu \nu }$ by

Thus $H_1=G_{\mathrm{lin}}$. Appendix app:coefficients gives a finite coefficient prescription.

Also,

with

The invariant metric Euler density is

It contains no derivatives of $g$.

By contrast, the scalar Euler density is

Equivalently,

where

Finally,

Proof. The formulas are XI-A :euler and :scalar-switch CheungXIA. Combining (eq:stressvariation) with $\delta q=\kappa \delta \gamma$ gives (eq:eexact); no metric integration by parts in the matter action is involved. Scalar integration by parts instead gives (eq:Ephiexact), and hence (eq:scalar-switch). The identity ${\partial }_{\mu }{T_0}^{\mu \nu }=(P_m\varphi ){\partial }^{\nu }\varphi$ gives (eq:sourcedivergence). All operations are completion-chart operations and transport by Lemma lem:transfer.

The action is fixed before its derivative is projected. The classical cubic and quartic Einstein coefficients supply $2H_2$ and $2H_3$. Because (eq:eexact) contains $1/\kappa$, a covariance check through order ${\kappa }^2$ must retain the required order-three metric curvature data. A premature order-two truncation of the pulled-back metric is not sufficient.

identities and tensor-density covariance

correct adjoint operator

For a symmetric contravariant density $u^{\mu \nu }$, define

The scaled active variations are

where

The backgrounds $\eta$, $g$, and ${\sigma }_{\epsilon }$ are held fixed in this active transformation.

Define

The exact invariant identity is

The gauge-fixed identity is instead

On $U$, ${\mathcal{B}}_{\nu }=0$, but the gauge-fixing contribution is not removed.

Proof. These are XI-A :noether, :noether, and :noether-gf CheungXIA. For a compact interior parameter, direct integration by parts gives

The inherited fixed-background action variation has coefficient $\kappa \int {\xi }^{\nu }{\mathcal{B}}_{\nu }{\mathrm{d}}^{4}x$. Comparison gives (eq:noether). The variation of the independent term (eq:gf) gives (eq:noethergf). The pairing transfers by Lemma lem:transfer. No new proof of the relational Route II theorem is asserted.

The raw divergence and the adjoint Noether operator are different:

The factor two comes from the leading flat part of (eq:N).

For a $\kappa$-independent scalar, write $e=e_0+\kappa e_1+{\kappa }^2{e}_2+\cdots$, and put

The retained coefficient audit is

where

These formulas retain the nonlinear adjoint contribution; a bare partial divergence is not the complete identity.

two rejected replacements

Replacing the full Einstein action by the Fierz–Pauli coefficient does not preserve the nonlinear Noether identity. The fixed gauge-fixing term is not invariant under ${\mathcal{G}}_P$.

Proof. For the first assertion, XI-A's order-one vacuum identity contains

For a tensor equal to $\psi (t)\eta$ near a point,

Deleting $H_2$ therefore leaves a generally nonzero term.

For the second assertion, take the same compact spatial parameter used in XI-A, $\xi =f{\partial }_1$, $0\ne f\in {C_c}^{\infty }(U)$. Then

A nonzero spacetime-compact homogeneous wave is excluded by Cauchy uniqueness, so $\square f\not\equiv 0$. These are the existing XI-A checks, now inside the retained regular chart CheungXIA.

No gauge algebra is recovered by imposing both compact spacetime support and $\square \xi =0$: that restriction would leave only the zero parameter.

tensor-density covariance

For an orientation-preserving diffeomorphism $F$, define

For $F\in {\mathcal{G}}_P$, put $q^{\prime }=F^{\ast }q$, ${\varphi }^{\prime }=F^{\ast }\varphi$, and ${\gamma }^{\prime }=(q^{\prime }-\eta )/\kappa$. Then

The formal identity retains its coefficientwise meaning through order ${\kappa }^2$ when projection is performed after the required coefficients have been retained. It is not valid in general with $e+e_{\mathrm{gf}}$ in place of $e$.

For a compact near-identity flow not required to preserve $g$, its metric covariance defect is

Proof. This is XI-A :covariance and :covariance-defect CheungXIA. The two tensor densities in (eq:eexact) are natural. Their switched matter terms differ only by $g-F^{\ast }g$, giving (eq:covariance-defect). The plateau support gives $F^{\ast }g=g$. The explicit gauge-fixing operator (eq:egf) retains the fixed background and fails the corresponding assertion, as already diagnosed in Restatement rest:rejections. Lemma lem:transfer supplies the local host reading.

The infinitesimal covariance defect involves ${\xi }^{\mu }{\partial }_{\mu }g$. The source-divergence defect involves $({\partial }_{\mu }g){T_0}^{\mu \nu }$. They are related background-switch effects, not identical formulas.

completion Einstein jet and its compact detector

equation space being tested

For a smooth free scalar ${\varphi }_0$, let ${\mathcal{J}}_g({\varphi }_0)$ consist of smooth coefficient families

such that the coefficients through order ${\kappa }^2$ of

vanish on the completion slab. The scalar satisfies $P_m{\varphi }_0=0$, and ${\varphi }_1,{\varphi }_2$ have the inherited zero Cauchy data at $\Sigma$. Metric Cauchy data are unrestricted.

This is the invariant Einstein equation, not the gauge-fixed Euler equation.

For clarity, the scalar coefficient equations include

These equations specify the inherited classical Cauchy problem. No inverse is constructed here.

The metric coefficients include

where

Thus the scalar solution correction is not confused with the metric correction to the stress density.

Nonemptiness of ${\mathcal{J}}_g({\varphi }_0)$ requires

In the subclass ${\gamma }_0=0$, this is already necessary for

Proof. This is XI-A :compatibility CheungXIA. For ${\gamma }_0=0$, take the divergence of (eq:sourcejet) and use ${\partial }_{\mu }{G_{\mathrm{lin}}}^{\mu \nu }=0$ and ${\partial }_{\mu }{T_0}^{\mu \nu }[{\varphi }_0]=0$. For arbitrary ${\gamma }_0$, use the order-one coefficient of (eq:noether). On a purported jet its left side vanishes; ${E_{\varphi }}^{[0]}=-P_m{\varphi }_0=0$, while ${{\mathcal{B}}_{\nu }}^{[0]}=({\partial }_{\mu }g){T_0}^{\mu }{}_{\nu }[{\varphi }_0]$. The same condition follows without restricting the leading vacuum metric data.

inherited empty fiber

Define

For every nonzero compact switch of the frozen class and every fixed $m\ge 0$,

The vacuum provides a nonempty fiber, so universal emptiness is false.

Proof. The inherited proof is XI-A :empty CheungXIA. For (eq:input),

The time component of (eq:compatibility) would require ${\partial }_{t}g/2=0$. A nonzero switch compactly supported in the slab interior cannot be independent of time. Hence compatibility fails.

For ${\varphi }_0=0$, the choice

solves the coefficient equations. A higher-order correction cannot cancel a nonzero order-one coefficient.

quantitative bump and its actual support

There is a relatively compact open region $B$ and a constant $c_{\mathrm{b}}>0$ with

Choose $f_{\mathrm{b}}\ge 0$ with

The retained XI-A detector is

If $\mathrm{supp}{f}_{\mathrm{b}}\subset D_{\ast }$, the nonzero pairing (eq:bump) transfers to the retained laboratory with the same value.

The detector is a switching-region detector: its support is not contained in the plateau $U$. If it does not fit the retained chart or shadow, the completion obstruction remains valid, but a report of its supplied value is not a locally acquired detector in that laboratory. Enlarging the shadow is not authorized.

Proof. Lemma lem:transfer preserves the compact pairing. On $U$, $g=1$ on a neighborhood and ${\partial }_{t}g=0$; thus a test supported there cannot have the strictly positive pairing (eq:bump). The remaining statements distinguish the domain of the analytical calculation from the location and access contract of its report.

Definition def:jet requires the equations on the completion slab. It is not the space obtained by requiring equations only on a plateau neighborhood. The obstruction proves no universal nonexistence theorem for plateau-local solutions.

Where the detecting switching region is in the chart, the same local compatibility identity excludes a jet satisfying the invariant equations on a neighborhood of that detector. This is the precise local consequence.

For the massless input, the existing XI-A conformal check is

Selecting $\gamma =\psi \eta$ does not change (eq:bump), which depends only on $g,{\varphi }_m,f_{\mathrm{b}}$. It therefore does not repair the sourced equation.

The other retained diagnostic is

It checks the linearization, not existence of a sourced jet.

The optional compact-scalar-data version of the witness uses a spatial cutoff equal to one on a ball containing the initial causal shadows of $\mathrm{supp}g$. Finite propagation makes its scalar solution agree with (eq:input) near $\mathrm{supp}g$, preserving the detector CheungXIA,BGP. This does not construct a gravitational propagator.

the gauge-fixed wave equation is different

For the source-response subclass, the quadratic gauge-fixed equation would be

Taking its divergence on the free scalar shell yields

For ${\varphi }_m$, the right side at $\nu =0$ is $-{\partial }_{t}g/4$, so $D{\gamma }_1$ cannot vanish identically.

Componentwise smooth Cauchy theory can solve (eq:wave) without a conserved source. That fact does not solve (eq:sourcejet). This is XI-A's distinction, later used in XI-B CheungXIA,CheungXIB.

only{ Equation (eq:quadraticoperator) specifies the signed quadratic operator. This paper supplies no gravitational retarded or advanced inverse, no two-point function, no ghost or auxiliary multiplet, no antifield, no tree, and no state.

XI-B is a downstream comparison, not a premise used to construct those objects here.}

The source obstruction is not a statement that Einstein theory fails. It is an incompatibility between the specified compact switch, the named free input, and the invariant equation space.

coverage and the limits of a report catalogue

The proposed catalogue records finitely many evaluations. It cannot replace an identity on an unrestricted smooth or distributional space. The following local statements give the coverage and invisibility discipline compared with Oai282-A CheungOai282.

Let $Z$ be a proved finite-dimensional residual space, and let

be a specified linear probe map. The conditions ${\ell }_j(r)=0$ for every $j$ force $r=0$ for every $r\in Z$ if and only if $P$ is injective.

For any fixed norm on $Z$, injectivity implies a finite constant $C_Z$ with

Neither the subspace membership $r\in Z$ nor injectivity follows from having printed a finite table.

Proof. The exact assertion is $\ker P=\{0\}$. For the quantitative assertion, the continuous function $r\mapsto {\Vert P(r)\Vert }_{\infty }$ has a positive minimum on the unit sphere of a nonzero finite-dimensional $Z$. Its reciprocal gives $C_Z$. The zero space is immediate.

Let $O\subset V$ be any nonempty open set and ${\ell }_1,\dots ,{\ell }_N$ any finite family of linear evaluations on smooth compactly supported symmetric tensors. There is a nonzero

such that

No assertion of a zero trace is required.

Proof. For $\psi \in {C_c}^{\infty }(O)$, define

It is symmetric and divergence free. If $S_{\psi }=0$, taking its trace gives $-3\square \psi =0$. A spacetime-compact homogeneous wave is zero by Cauchy uniqueness, so $\psi =0$.

Choose $N+1$ nonzero bump functions with mutually disjoint compact supports in $O$. Their tensors $S_{{\psi }_a}$ are nonzero and linearly independent. The $N$ evaluations on their span have a nonzero kernel. A nonzero tensor in that kernel has all the required properties.

This establishes the finite-probe limitation locally. It does not append a new stress source to the action, construct an Einstein solution, or verify OpenAI Family 282.

The independent non-implication from record conservation to Einstein dynamics is the existing I-B :not-source and X-A-ST :noeinstein boundary CheungIB,CheungXAST. No third geometric counterexample is needed here.

complete Enter report interface

finite supplied catalogue

Before execution, freeze all evaluation configurations, tests, units, source data, and approximation enclosures. A possible catalogue consists of the following entries.

Vector entries are flattened into finitely many real components. Every entry is an ordinary real number. A field, tensor density, equation space, and truth value are not identified with one another by this flattening.

An approximation enclosure must be justified by its own mathematical calculation or supplied certificate. A channel-distance bound does not validate that enclosure.

clock readout

Choose

with orthonormal calibration basis $\{|{\tau }_j⟩:0\le j<D\}$. Let $S$ contain every unmeasured finite register. Set

Unused clock codes have a predeclared no-task or failure interpretation; no branch is omitted.

The channel

is CPTP and equals

For every untouched reference $E$,